How many orbitals in an atom can have the designation n = 3, l = 2?

• A. 5
• B. 3
• C. 1
• D. 7

Answer: (A)

Verified step by step guidance

1. Understand the quantum numbers involved: \(n\) is the principal quantum number, which indicates the energy level or shell, and \(l\) is the azimuthal (angular momentum) quantum number, which indicates the subshell or shape of the orbital.
2. Recall that for a given \(l\), the magnetic quantum number \(m_l\) can take integer values from \(-l\) to \(+l\), including zero. This means the number of possible \(m_l\) values (and thus orbitals) for a given \(l\) is \$2l + 1$.
3. Given \(l = 2\), calculate the number of orbitals by applying the formula: number of orbitals = \$2l + 1$.
4. Substitute \(l = 2\) into the formula: number of orbitals = \$2 \times 2 + 1$.
5. Conclude that the number of orbitals with \(n = 3\) and \(l = 2\) corresponds to the number of \(d\) orbitals in the third shell, which is the result from the previous calculation.